c
TU Delft, The Netherlands, 2006
ERROR ESTIMATION AND ADJOINT-BASED
ADAPTATION IN AERODYNAMICS
Ralf Hartmann∗
∗ German Aerospace Center (DLR)
Institute of Aerodynamics and Flow Technology (AS) Lilienthalplatz 7, 38108 Braunschweig
e-mail: Ralf.Hartmann@dlr.de web page: http://www.dlr.de/as
Key words: Error estimation, Adjoint-based refinement, Discontinuous Galerkin dis-cretization, Compressible flows, Aerodynamics
Abstract. In this article we give an overview of recent developments in error estima-tion and in residual-based and goal-oriented (adjoint-based) adaptaestima-tion for Discontinuous Galerkin discretizations of sub- and supersonic viscous compressible flows. We also give an outlook on the planned continuation of this research in the EU project ADIGMA.
1 INTRODUCTION
In aerodynamical computations like compressible flows around airfoils, much emphasis is placed on the accurate approximation of specific target quantities J (·), in particular, the aerodynamical force coefficients like the pressure induced as well as the viscous stress induced drag, lift and moment coefficients, respectively. While local mesh refinement is required for obtaining reasonably accurate results in applications, the goal of the adap-tive refinement is either to compute these coefficients as accurate as possible within given computing resources or to compute these coefficients up to a given tolerance with the min-imum computing resources required. In both cases a goal-oriented refinement is needed, i.e. an adaptive refinement strategy specifically targeted to the efficient computation of the quantities of interest. Furthermore, in the latter case, an estimate is required of how accurate the force coefficients are approximated, i.e. an a posteriori error estimate is required of the error of the numerical solution measured in terms of the quantity of interest.
be used for goal-oriented (adjoint-based) refinement specifically tailored to the efficient computation of the quantities of interest.
The approach of a posteriori error estimation and adaptivity in finite element meth-ods has been developed in [1] and applied to various kinds of problems, see the survey article [2]. In [4], this approach has been developed for the discontinuous Galerkin dis-cretization of scalar hyperbolic problems. Then, in the series of publications, [5, 7, 8], it has been extended to the two–dimensional compressible Euler equations, where a variety of problems have been considered, including the Ringleb flow problem, supersonic flow past a wedge, inviscid flows through a nozzle, and inviscid sub-, trans- and supersonic flows around different airfoil geometries; finally, in [9] and [10], this approach has been extended to the two–dimensional compressible Navier-Stokes equations and applied to subsonic viscous compressible flows around simple airfoil geometries. [6] gives the exten-sion of this approach to viscous compressible flows including shocks, like supersonic flows, for example.
In this publication we give an overview of recent developments in the a posteriori error estimation as well as residual-based and goal-oriented (adjoint-based) adaptation for Discontinuous Galerkin discretizations of sub- and supersonic compressible flows. First we present an overview of the general theoretical framework of duality-based (adjoint-based) a posteriori error estimation in Section 2. Then, we introduce the Discontinuous Galerkin (DG) discretization of the compressible Euler equations in Section 3 and the Interior Penalty DG discretization of the compressible Navier-Stokes equations in Section 4 and give the corresponding residual-based and adjoint-based refinement indicators used for adaptive mesh refinement. Then, in Section 5 we present some numerical examples highlighting the quality of the a posteriori error estimation and the advantage of using adjoint-based mesh refinement over residual-based mesh refinement. In the concluding Section 6 we give an outlook on the further development of these algorithms for the use in aerodynamical applications as planned in the EU project ADIGMA.
2 A POSTERIORI ERROR ESTIMATION
In this section we give an overview of the general theoretical framework of duality-based (adjoint-based) a posteriori error estimation developed by C. Johnson and R. Rannacher and their collaborators, [1, 2, 3] and the references cited therein.
Let V be a Hilbert space. Further, we write N (·, ·) to denote a semi-linear form (nonlinear in its first argument, but linear in its second), with derivative Nu0(·; ·, ·). We suppose that u is the unique solution to the variational problem: find u in V such that
N (u, v) = 0 ∀v ∈ V. (1)
In order to construct a Galerkin approximation to this problem, we consider a sequence of finite–dimensional spaces {Vh}, parameterized by the positive discretization parameter
h; for the sake of simplicity we suppose that Vh ⊂ V for each h. For the purposes of this
functions on a partition, of granularity h, of the computational domain. The Galerkin approximation uh of u is then sought in Vh as the solution of the finite–dimensional
problem
N (uh, vh) = 0 ∀vh ∈ Vh. (2)
For simplicity of presentation, we assume that Vh is a suitably chosen finite element
space to ensure the existence of a unique solution uh to (2). Furthermore, we assume that
the discretization (2) is consistent, i.e. the exact solution u of (1) satisfies the discrete problem, i.e.
N (u, vh) = 0 ∀vh ∈ Vh. (3)
Combining (2) and (3) we obtain the so-called Galerkin orthogonality of the discretization: N (u, vh) − N (uh, vh) = 0 ∀vh ∈ Vh, (4)
which will be a key ingredient in the following a posteriori error analysis.
Assuming that the functional of interest J (·) is differentiable with derivative J0[w](·) at some w in V, we write ¯J (u, uh; ·) to denote the mean value linearization of J (·) defined
by
¯
J (u, uh; u − uh) = J (u) − J (uh) =
Z 1
0
J0[θu + (1 − θ)uh](u − uh) dθ, (5)
Analogously, we write M(u, uh; ·, ·) to denote the mean–value linearization of N (·, ·) given
by
M(u, uh; u − uh, v) = N (u, v) − N (uh, v)
= Z 1
0
Nu0[θu + (1 − θ)uh](u − uh, v) dθ (6)
for all v in V. We now introduce the following dual problem (or adjoint problem): find z ∈ V such that
M(u, uh; w, z) = ¯J (u, uh; w) ∀w ∈ V. (7)
For the proceeding error analysis, we assume that the dual problem (7) is well–posed. Under this assumption, employing the Galerkin orthogonality property (4) we deduce the following error representation formula:
J (u) − J (uh) = J (u, u¯ h; u − uh)
= M(u, uh; u − uh, z)
= M(u, uh; u − uh, z − zh)
for all zh in the finite element space Vh. Let us now decompose the right–hand side of
(8) as a summation of local error indicators ηκ over the elements κ in the computational
mesh Th, i.e. we write
J (u) − J (uh) = − N (uh, z − zh) ≡ X κ∈Th ηκ =X κ∈Th Z κ R(uh)(z − zh) dx + Z ∂κ r(uh)(z − zh) ds , (9)
where R(uh) and r(uh) are element and face residuals of the discretization (2).
As in most cases the exact solution z to the dual problem (7) is not known, it is approximated numerically. However, (7) includes the unkown exact solution u to the primal problem. Thus, in order to approximate the dual solution z, we must replace u in (7) by a suitable approximations. The linearizations leading to M(u, uh; ·, ·) and
¯
J (u, uh; ·) are performed about uh, resulting in Nu0[uh](·, ·) and J0[uh](·), respectively.
The linearized dual problem: find ˆz ∈ V such that
Nu0[uh](w, ˆz) = J0[uh](w) ∀w ∈ V, (10)
is then discretized to yield following approximate dual problem: find ˆzh ∈ ˆVh such that
ˆ
Nu0[uh](wh, ˆzh) = J0[uh](wh) ∀wh ∈ ˆVh. (11)
Replacing the dual solution z in (9) by its approximation ˆzh results in following
approxi-mate error representation formula
J (u) − J (uh) ≈ −N (uh, ˆzh− zh) ≡
X
κ∈Th
ˆ
ηκ. (12)
We note that the error introduced into the error representation through this replacement consists of the linearization and the discretization error of the dual problem, see [6] for a more detailed discussion. Furthermore, we note that the indicators ˆηκ are used for
goal-oriented (adjoint-based) refinement. Finally, based on the error representation (9) given in terms of element and face residuals, R(uh) and r(uh), respectively, we can derive
3 THE COMPRESSIBLE EULER EQUATIONS
We consider the two-dimensional steady state compressible Euler equations of gas dynamics given by
∇ · Fc(u) = 0 in Ω, (13)
where Ω ∈ R2 is an open bounded domain, Fc(u) = (fc
1(u), f2c(u)), and the vector of
conservative variables u and the convective fluxes fic, i = 1, 2, are defined by
u = ρ ρv1 ρv2 ρE , f1c(u) = ρv1 ρv21+ p ρv1v2 ρHv1 and f2c(u) = ρv2 ρv1v2 ρv2 2+ p ρHv2 , (14)
where ρ, v = (v1, v2)T, p and E denote the density, velocity vector, pressure and specific
total energy, respectively. Additionally, H is the total enthalpy given by H = E + p ρ = e + 1 2v 2+ p ρ, (15)
where e is the specific static internal energy, and the pressure is determined by the equation of state of an ideal gas, p = (γ−1)ρe, where γ = cp/cvis the ratio of specific heat capacities
at constant pressure, cp, and constant volume, cv; for dry air, γ = 1.4.
Given a subdivision of Ω into shape-regular meshes Th = {κ} consisting of quadrilateral
elements κ, and mappings σκ, κ ∈ Thwith κ = σκ(ˆκ) where ˆκ is the reference (unit) square,
we define the finite element space Vph of discontinuous piecewise vector-valued polynomial functions of degree p ≥ 0 by
Vhp = {vh ∈ [L2(Ω)]m : vh|κ◦ σκ ∈ [Qp(ˆκ)]m}, (16)
where Qp(ˆκ) denotes the space of tensor product polynomials of degree p ≥ 0. Suppose
that v|κ ∈ [H1(κ)] m
for each κ ∈ Th. Given an element κ ∈ Th and neighoring element
κ0 ∈ Th with e = ∂κ ∩ κ0 6= 0, by v±κ (or v± for short) we denote the traces of v taken from
within the interior of κ and κ0, respectively. The discontinuous Galerkin discretization of degree p ≥ 0 of problem (13) is given by: Find uh ∈ Vph such that
N (uh, vh) ≡ X κ∈Th − Z κ Fc(uh) : ∇vhdx + Z ∂κ H(u+h, u−h, n) v+h ds = 0 (17)
for all vh ∈ Vph, where H(·, ·, ·) is a consistent and conservative numerical flux function,
see [8] for more details. Substituting the semilinear form N (uh, vh) given in (17) into the
error representation formula (9) and using integration by parts, we see that the adjoint-based indicators ηκ in (9) are defined by
and the element and face residuals in (9) are given by
R(uh) = −∇ · Fc(uh), r(uh) = Fc(u+h) · n − H(u + h, u
−
h, n), (19)
respectively. Finally, supposing that z ∈ [H1(Ω)]4, and there is a constant Cstab such that
kzkH1(Ω) ≤ Cstab, we can derive, see [8, 11], following residual-based indicators
ηres
κ = khR(uh)kL2(κ)+ kh
1/2r(u
h)kL2(∂κ). (20)
4 THE COMPRESSIBLE NAVIER-STOKES EQUATIONS
We consider the two–dimensional steady state compressible Navier-Stokes equations. Like in Section 3, ρ, v = (v1, v2)T, p and E denote the density, velocity vector, pressure
and specific total energy, respectively. Furthermore, T denotes the temperature. The equations of motion are given by
∇ · (Fc(u) − Fv(u, ∇u)) ≡ ∂
∂xi
fic(u) − ∂ ∂xi
fiv(u, ∇u) = 0 in Ω. (21) The vector of conservative variables u and the convective fluxes fic, i = 1, 2, are given by (14). Furthermore, the viscous fluxes fv
i , i = 1, 2, are defined by f1v(u, ∇u) = 0 τ11 τ21 τ1jvj + KTx1
and f2v(u, ∇u) = 0 τ12 τ22 τ2jvj+ KTx2 , (22)
respectively, where K is the thermal conductivity coefficient. Finally, the viscous stress tensor is defined by
τ = µ ∇v + (∇v)T − 2
3(∇ · v)I , (23)
where µ is the dynamic viscosity coefficient, and the temperature T is given by e = cvT ;
thus
KT = µγP r E −12v2 , (24)
where P r = 0.72 is the Prandtl number. Finally, we note that the viscous flux Fv(u, ∇u)
is homogeneous with respect to the gradient of conservative variables ∇u, i.e. fiv(u, ∇u) = Gij(u)∂u/∂xj, i = 1, 2, where G denotes the homogeneity tensor and is given by Gij(u) =
∂fv
i (u, ∇u)/∂uxj, for i, j = 1, 2.
In addition to the notation introduced in Section 3, we now define average and jump operators. To this end, let κ+ and κ− be two adjacent elements of T
h and x be an
arbitrary point on the interior edge e = ∂κ+ ∩ ∂κ− ⊂ Γ
I, where ΓI denotes the union
of all interior edges of Th. Moreover, let v and τ be vector- and matrix-valued functions,
at x ∈ e by {{v}} = (v++ v−)/2 and {{τ }} = (τ++ τ−)/2. Similarly, the jump at x ∈ e
is given by [[v]] = v+⊗ nκ+ + v−⊗ nκ−. On a boundary edge e ⊂ Γ, we set {{v}} = v,
{{τ }} = τ and [[v]] = v ⊗ n. For matrices σ, τ ∈ Rm×n, m, n ≥ 1, we use the standard
notation σ : τ = Pm
k=1
Pn
l=1 σklτkl; additionally, for vectors v ∈ R m
, w ∈ Rn, the matrix v ⊗ w ∈ Rm×n is defined by (v ⊗ w)
kl= vkwl.
The interior penalty discontinuous Galerkin discretization of the compressible Navier-Stokes equations is given, [9], by
N (uh, vh) ≡ − Z Ω Fc(u h) : ∇hvhdx + X κ∈Th Z ∂κ\Γ H(u+ h, u − h, nκ) · v+h ds + Z Ω Fv(uh, ∇huh) : ∇hvhdx − Z ΓI {{Fv(uh, ∇huh)}} : [[vh]] ds − Z ΓI {{ GT i1∂hvh/∂xi, Gi2T∂hvh/∂xi}} : [[uh]] ds + Z ΓI δ[[uh]] : [[vh]] ds + NΓ(uh, vh), (25)
where NΓ(uh, vh) includes all boundary terms, see [9] for more details.
Substituting the semilinear form N (uh, vh) given in (25) into the error representation
formula (9) and, again, using integration by parts, we see that the adjoint-based indicators ηκ in (9) are given by ηκ = Z κ (−∇ · Fc(uh) + ∇ · Fv(uh, ∇uh)) · (z − zh) dx + Z ∂κ\Γ (Fc(uh) · nκ− H(u+h, u−h, nκ)) · (z − zh)+ds + 1 2 Z ∂κ\Γ G>i1∂h(z − zh)/∂xi, G>i2∂h(z − zh)/∂xi : [[uh]] ds − 1 2 Z ∂κ\Γ [[[[[[Fv(uh, ∇uh)]]]]]] · (z − zh)+ds − Z ∂κ\Γ δ[[uh]] : (z − zh)+⊗ nκds + η∂κ∩Γ,
where η∂κ∩Γ includes the residual contributions of the boundary terms NΓ(uh, vh), see [10]
for more detail. Finally, supposing that z ∈ [Hs(Ω)]4, 2 ≤ s ≤ p + 1, and that there is a constant Cstab such that kzkHs(Ω) ≤ Cstab, we can derive, cf. [10], following residual-based
indicators ηres κ =kh s κR(uh)kL2(κ)+ kh s−1/2 κ (F c(u h) · nκ− H(u+h, u−h, nκ))kL2(∂κ\Γ) + khs−3/2κ G·j[[uh]]jkL2(∂κ\Γ)+ kh s−1/2 κ [[[[[[F v(u h, ∇uh)]]]]]]kL2(∂κ\Γ) + khs−1/2κ δ u+h − u−h kL2(∂κ\Γ)+ η res ∂κ∩Γ, (26)
where R(uh) = −∇ · Fc(uh) + ∇ · Fv(uh, ∇uh) denotes the element residual, and η∂κ∩Γres
Figure 1: Mach isolines of the M = 0.5, Re = 5000, α = 0◦ flow around the NACA0012 airfoil.
5 NUMERICAL EXAMPLES
In this Section we give two numerical examples demonstrating that the approximate error representation −N (uh, ˆzh − zh) =
P
κ∈Thηˆκ, cf. (12), which was derived from the
(exact) error representation (9) by replacing the dual solution z by an approximate dual solution ˆzh, gives a good approximation to the true error measured in terms of the specific
target quantity J (u). Furthermore, we show that using the approximate error represen-tation, an improved value of the target functional, namely, ˜J (uh) = J (uh) +
P
κ∈Thηˆκ
can be obtained. Finally, we highlight the advantages of designing an adaptive finite ele-ment algorithm based on the adjoint-based indicators ˆηκ, in comparison to residual-based
indicators ηres
κ , which do not require the solution of an adjoint problem.
First, we consider the example, see [10], of a subsonic viscous flow (M = 0.5, Re = 5000, α = 0◦) around the NACA0012 airfoil with an adiabatic no-slip boundary condition imposed on the profile, see Figure 1. In Table 1 we demonstrate the performance of the adaptive algorithm for the numerical approximation of the viscous drag coefficient J (u) = Jcdf(u) = 2 ¯ l ¯ρ|¯v|2 R S(τ n) · ψdds with ψd = (1, 0)
>, when employing the
adjoint-based indicators ˆηκ. Here, we show the number of elements and degrees of freedom
(DoF) in Vh, the true error in the functional J (u) − J (uh) based on a reference value
Jcdf(u) ≈ 0.032535, the computed error representation formula and its effectivity index
θ =P
κ∈Thηˆκ/(J (u) − J (uh)). We see that initially on very coarse meshes the quality of
the computed error representation formula P
κ∈Thηˆκ is rather poor, in the sense that θ
noticeably differs from one; however, as the mesh is refined, we observe that the effectivity indices θ slowly tend towards unity.
In Figure 2 we compare the true error in the computed target functional Jcdf(·) using a
mesh refinement strategy based on the residual-based indicators ηres
κ and a mesh refinement
strategy based on the adjoint-based indicators ˆηκ. Here, we clearly observe the superiority
of employing the adjoint-based indicator; on the final mesh, the true error in the computed target functional is almost 2 orders of magnitude smaller than |Jcdf(u)−Jcdf(uh)| computed
on the sequence of meshes produced using ηres
κ . Moreover, here we also show the error in
the improved value of the viscous drag coefficient, i.e. |Jcdf(u) − ˜Jcdf(uh)|; in this case, we
clearly see that this error is of higher–order than the baseline error |Jcdf(u) − Jcdf(uh)|.
Table 1: M = 0.5, Re = 5000, α = 0◦ flow around the NACA0012 airfoil: Adaptive algorithm for the
numerical approximation of cdf based on employing adjoint-based indicator ˆηκ.
Elements DoF J (u) − J (uh)
P κ∈Thηˆκ θ 3072 49152 -1.839e-02 -1.274e-02 0.69 4962 79392 -3.680e-03 -3.239e-03 0.88 8028 128448 -8.246e-04 -7.596e-04 0.92 13446 215136 -1.773e-04 -1.680e-04 0.95 21750 348000 -4.444e-05 -4.258e-05 0.96 35118 561888 -1.624e-05 -1.626e-05 1.00 0.032 0.034 0.036 0.038 0.04 0.042 0.044 0.046 0.048 0.05 0.052 3000 10000 30000 cdf cells reference cdf = 0.032535 cdf value for ref. by ind. eta^(II) cdf value for ref. by ind. eta^(I) improved cdf value for ref. by ind. eta^(I)
1e-06 1e-05 0.0001 0.001 0.01 3000 10000 30000 error in cdf cells
error of cdf value for ref. by ind. eta^(II) error of cdf value for ref. by ind. eta^(I) error of improved cdf value for ref. by ind. eta^(I)
(a) (b)
Figure 2: M = 0.5, Re = 5000, α = 0◦ flow around the NACA0012 airfoil: (a) Computed values of
cdf based on employing the adjoint-based indicator (eta(I)) and the residual-based indicator (eta(II)),
together with the improved value; (b) Convergence of the error in these quantities.
of magnitude smaller than the corresponding quantity computed with the adjoint-based indicator. We also point out that after just one mesh refinement step, the improved value
˜
Jcdf(uh) computed on the mesh refined using the adjoint-based indicator is more accurate
than the corresponding value Jcdf(uh) computed on the finest mesh designed on the basis
of employing the residual-based indicator.
In a second example, cf. [6], we consider a supersonic horizontal viscous flow at M = 1.2 and Re = 1000 around the NACA0012 airfoil, with an adiabatic no-slip boundary condition imposed on the profile, see Figure 3. In order to avoid overshoots near the bow shock, we add a consistent shock-capturing term, see [6], to the discretization (25). In this example, we now consider the approximation of the pressure induced drag, cdp, i.e.
the target quantity is J (u) = Jcdp(u) =
2 ¯ l ¯ρ|¯v|2 R Sp (n · ψd) ds, with ψd= (1, 0) >.
In Table 2 we collect the data of the adaptive refinement algorithm when the adjoint-based indicators are employed. Again, we see that from the second mesh onwards, the approximate error representationP
κηˆκ is very close to the true error J (u) − J (uh) based
(a) (b)
Figure 3: M = 1.2, Re = 1000, α = 0◦ flow around the NACA0012 airfoil: (a) Mach isolines and (b)
density isolines.
# el. # DoFs J (u) − J (uh) Pκηˆκ θ 768 12288 -1.363e-02 -6.312e-03 0.46 1260 20160 -3.203e-03 -2.995e-03 0.94 2154 34464 -4.844e-04 -5.368e-04 1.11 3570 57120 -3.474e-04 -3.333e-04 0.96 6021 96336 -1.835e-04 -1.856e-04 1.01 10038 160608 -1.644e-04 -1.653e-04 1.01
Table 2: Viscous M = 1.2, Re = 1000, α = 0◦flow around the NACA0012 airfoil: Adaptive algorithm for
the accurate approximation of cdp.
0.1 0.102 0.104 0.106 0.108 0.11 0.112 0.114 0.116 1000 10000 cdp cells
cdp value for ref. by ind. eta^(II) cdp value for ref. by ind. eta^(I) improved cdp value for ref. by ind. eta^(I) reference cd = 0.10109 1e-05 0.0001 0.001 0.01 0.1 1000 10000 error in cdp cells
error of cdp value for ref. by ind. eta^(II) error of cdp value for ref. by ind. eta^(I) error of improved cdp value for ref. by ind. eta^(I)
(a) (b)
Figure 4: M = 1.2, Re = 1000, α = 0◦ flow around the NACA0012 airfoil: (a) Computed values of
cdf based on employing the adjoint-based indicator (eta(I)) and the residual-based indicator (eta(II)),
-8 -4 0 4 8 -4 0 4 8 -8 -4 0 4 8 -4 0 4 8 (a) (b)
Figure 5: M = 1.2, Re = 1000, α = 0◦flow around the NACA0012 airfoil: (a) residual-based refined mesh
of 17670 elements with 282720 degrees of freedom and |Jcdp(u) − Jcdp(uh)| = 1.9 · 10
−3; (b) adjoint-based
refined mesh for cdp: mesh of 10038 elements with 160608 degrees of freedom and |Jcdp(u) − Jcdp(uh)| =
1.6 · 10−4.
In Figure 4 we compare the true error in the target quantity for the two mesh refinement strategies based on the adjoint-based indicator ˆηκ and on the residual-based indicator ηκres,
respectively. We see, that on the first three refinement steps when employing the residual-based indicator the accuracy in the target quantity is hardly improved. In contrast to that, when using adjoint-based indicators, the error decreases significantly faster, being a factor of more than three smaller already after the second refinement step than the error on the finest residual-based refined mesh. Furthermore, in Figure 4 we see, that the improved values, ˜J (uh), are significantly more accurate than the (baseline) J (uh) values,
and even show a higher rate of convergence, see [6] for a more detailed discussion. The large difference in the performance, see Figure 4, of the adjoint-based indicators and the residual-based indicators in producing adaptively refined meshes for the accurate approximation of the target quantity cdp, is due to the very different parts of the
-3 -2 -1 0 1 2 3 -2 -1 0 1 2 -3 -2 -1 0 1 2 3 -2 -1 0 1 2 (a) (b)
Figure 6: Viscous flow at M = 1.2, Re = 1000, α = 0◦ around the NACA0012 airfoil: (a) Sonic isolines
of the flow solution; (b) isolines of the ˆz1 component of the computed adjoint solution ˆz.
trailing edge are not resolved and there is no refinement in the wake of the flow beyond three chord lengths behind the profile. Instead, the refinement of the mesh is concen-trated near the leading edge of the profile and in the boundary layer of the flow. All other parts of the computational domain are recognized by the adjoint-based indicator to be of minor importance for the accuracy of the cdp target quantity. In fact, the dual (adjoint)
solution, see Figure 6, includes the crucial information concerning which local residuals contribute to the error in the target quantity and to what extent. Herewith, it offers all necessary information of error transport and accumulation. Finally, the adjoint-based indicators mark only those parts of the domain for refinement where residuals of the flow solution significantly contribute to the error of the target quantity, i.e. all parts which are important for the accurate approximation of the target quantity.
6 CONCLUSION AND OUTLOOK
corre-sponding residual-based and adjoint-based refinement indicators used for adaptive mesh refinement. We have demonstrated for a subsonic and a supersonic viscous compressible flow that a reliable error estimation is obtained with respect to specific target quantities like aerodynamical force coefficients. Furthermore, we have shown that the error estima-tion can be used to obtain improved values (of higher order convergence) for the target quantities. Finally, we demonstrated that mesh refinement using adjoint-based indicators produces meshes which are specifically tailored to the efficient computation of the quan-tities of interest. It has been shown for a sub- and a supersonic viscous compressible flow that the adjoint-based mesh refinement leads to a several orders of magnitude improved accuracy compared to residual-based mesh refinement for the same number of points.
The results show that there is an enormous potential in the presented methods for im-proving the efficiency and reliability of aerodynamical simulations. However, significant effort will be required to make these methods usable in large-scale aerodynamical applica-tions. To this end, in the context of the EU project ADIGMA (“Adaptive Higher-Order Variational Methods for Aerodynamic Applications in Industry”) it is planned to extend these methods to turbulent compressible high Reynolds flows on complex 3d geometries, as is required for exploitation in industrial applications.
7 ACKNOWLEGMENTS
This work has been supported by the President’s Initiative and Networking Fund of the Helmholtz Association of German Research Centres.
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